MIT 2.111/8.411/6.898/18.435 Quantum Information Science I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Sam Ocko
Fall, 2010
December 1, 2010
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
November 22, 2010
Problem Set # 10 Solutions
Problem Set #10
(
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
November 16, 2010
Problem Set #9 Solution
(due in class, TUESDAY 23-Nov-10)
P1: (Feynmans Hamiltonian model for QC) Read the paper Quantum Mechanical Computer
1
Id: hw2.tex,v 1.4 2009/02/09 04:31:40 ike Exp
MIT 2.111/8.411/6.898/18.435 Quantum Information Science I
Sam Ocko
Fall, 2010
November 15, 2010
Problem Set # 8 Solutions
1. (a) The eigenvectors of S1 S2 are |11 , |10 , |10 +|01 , |10 |01 having eigenvalu
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
October 28, 2010
Problem Set #7 Solution
(due in class, 04-Nov-10)
P1: (Continuous time Grover with noise) Consider a continuous time Grover algorithm on n qu
MIT 2.111/8.411/6.898/18.435 Quantum Information Science I
Sam Ocko
Fall, 2010
October 15, 2010
Problem Set # 5 Solutions
1. Most unitary transforms are hard to approximate.
(a) We are dealing with boolean functions that take n bits as input and output n
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
September 30, 2010
Problem Set #4 Solution
(due in class, 07-Oct-10)
1. Measurement in the Bell basis Show that the circuit
H
_ _ _ _ _ _ L
_ _ _ _ _ _
_ _ _
MIT 2.111/8.411/6.898/18.435 Quantum Information Science I
Sam Ocko
Fall, 2010
September 29, 2010
Problem Set # 3 Solutions
1. Measurements and Uncertainty
(a) The expectation value of M on state | will be m, the standard deviation will be 0.
(b) Measurin
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
September 16, 2010
Problem Set #2 Solution
1. Density matrices. A density matrix (also sometimes known as a density operator) is a representation
of statistic
1
Id: hw2.tex,v 1.4 2009/02/09 04:31:40 ike Exp
MIT 2.111/8.411/6.898/18.435 Quantum Information Science I
Sam Ocko
Problem Set #1 Solutions
1. (a) The eigenvectors of
0 I
10
01
are
1
0
,
0
1
Both of which have eigenvalue 1.
(b) The eigenvectors of
1 x X
18.435/2.111 Homework # 7 Solutions
If you have the POVM with three elements:
1
3
1
12
1
12
1
4
1
12
1
3
1
12
,
1
4
,
1
3
0
0
1
2
,
you can notice that the rst two elements have rank 1 (they can be expressed as
| v v | for some | v , and the third has r
18.435/2.111 Homework # 5 Solutions
1:
We start with the system in the state
|0 |u
After the Hadamard gate on the rst bit, we obtain
1
(| 0 + | 1 ) | u
2
After the control U gate, we get
1
(| 0 + e2i | 1 ) | u
2
Now, after the second Hadamard gate, we o
18.435/2.111 Homework # 4 Solutions
Problem 1: The trick here is to notice that controlled phase gates are symmetric: a
controlled phase from qubit i to qubit j is the same as a controlled phase from qubit
j to qubit i. With this fact, and the fact that a
18.435/2.111 Homework # 2 Solutions
1: First, notice that RT = T R.
1
The state | = 3 (| 00 + | 11 + | 22 ) works. We want to say that Ri T j I |
is orthogonal to Ri T j I | if i = i or j = j . Since RT commute at the cost of
a phase, we can do this if w
18.435/2.111 Homework # 1 Solutions
1a: I dont know any way to do this except multiply the whole thing out.
First,
cos ei sin
jx x + jy y + jz z =
.
i
e sin cos
Now, applying this to the vector
cos
2
ei sin
2
gives the vector
cos cos + sin sin 2
2
e (